Let be a
-algebra, and let
be a subset of its universe. The subset of
generated by
is defined to be the least subset of the universe which contains
and is closed under the multiplication operation of the algebra. By associativity, this subset is the same thing as the image of the multiplication operation over the set
. The following theorem shows that the elements generated by
can be generated by using only the basic operations that were already used in the characterisation theorem from the parent page.
Theorem. Let be a
-algebra with finite universe. For every subset
of the universe, the subset generated by
is the least subset that contains
and is closed under binary product, shuffle, as well as the powers
and
.
The proof of this theorem is similar to the one in the parent page.
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